Stats: Multiplication Rule & Counting Rule Confusion

[shawnz1102]

Hi everyone, I have trouble depicting the difference between these two problems. The first problem uses the multiplication rule to solve while the second problem uses the counting rules (combination) to solve. To me, it seems like both problems could be solved using the multiplication rule so I don't understand why the second problem is solved using combination...

1.) Three cards are drawn from an ordinary deck and not replaced. Find the probability of these: a) getting 3 jacks.

Solution: P(3 jacks) = 4/52 * 3/51 * 2/50 = 1/5525

2.) Find the probability of getting 4 aces when 5 cards are drawn from an ordinary deck of cards.

Solution: P(4 aces) = 4C4 * 48 / 52C5 = 1*48/2,598,960 = 1/54,145

If I reword the second problem, I could reword it as the following:

Five cards are drawn from an ordinary deck. Find the probability of these: a) getting 4 aces.

Wouldn't this make it the exact same as problem #1, meaning I could solve #2 the same way I solved #1?

 

[mathman]

If you do it right. Remember there is one free card which can be anything. In addition as you draw the five cards in sequence, the free card may be in any position, so you have to add up five cases (actually multiply one case by 5, since the probabilities are all the same).